Digital Communication Systems


 Elfrieda Hawkins
 3 years ago
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1 Digital Communication Sytem The term digital communication cover a broad area of communication technique, including digital tranmiion and digital radio. Digital tranmiion, i the tranmitted of digital pule between two or more point in a communication ytem. Digital radio, i the tranmitted of digital modulated analog carrier between two or more point in a communication ytem. Why Digital There are many reaon The primary advantage i the eae with which digital ignal, compared to analog ignal, are regenerative. The hape of the waveform i affected by two mechanim: (1) A all the tranmiion line and circuit have ome nonideal tranfer function, there i a ditorting effect on the ideal pule. () Unwanted electrical noie or other interference further ditort the pule waveform. Both of thee mechanim caue the pule hape to degrade a a function of ditance. During the time that the tranmitted pule can till be reliably identified, the pule i thu regenerated. The circuit that perform thi function at regular interval along a tranmiion ytem are called regenerative repeater. 1
2 Digital circuit are le ubject to ditortion and interference than analog circuit. Digital circuit are more reliable and can be produced at lower cot than analog circuit. Alo, digital hardware lend itelf to more flexible implementation than analog hardware. Digital technique lend themelve naturally to ignal proceing function that protect againt interference and jamming. Much data communication i computer to computer, or digital intrument or terminal to computer. Such digital termination are naturally bet erved by digital link. Communication Sytem Model Generally, there are two type for communication ytem model, baeband model and paband model. In baeband model, the pectrum of ignal from zero to ome frequency (i.e. carrier frequency=0). For tranmiion of baeband ignal by a digital communication ytem, the information i formatted o that it i repreented by digital ymbol. Then, pule waveform are aigned that repreented thee ymbol. Thi tep referred to a pule modulation or baeband modulation. Thee waveform can be tranmitted over a cable. Baeband ignal alo called lowpa ignal. In paband (or bandpa) ignal, the ignal ha a pectral magnitude that i nonzero for frequency in ome band concentrate about a frequency f f and negligible elewhere, where f C c i the carrier frequency need to be much greater than zero. For radio
3 tranmiion the carrier i covered to an electromagnetic (EM) filed for propagation to deired detination. Multiplexing Multiplexing i the tranmiion of information (either voice or data) from more than one ource to more than one detination on the ame tranmiion medium. Two mot common method are ued, frequency diviion multiplexing (FDM) and time diviion multiplexing (TDM). FDM In FDM multiple ource that originally occupied the ame frequency pectrum are each converted to a different frequency band and tranmitted imultaneouly over a ingle tranmiion medium. FDM i an analog multiplexing cheme. Figure below how the frequencytime plane. 3
4 If two input ignal to a mixer are inuoid with frequencie f A and f B, the mixing or multiplication will yield new um and difference frequencie at f A+B and f AB. Equation below decribe the effect of the mixer. 1 coacob [co( A B) co( A B)] A imple FDM example with three tranlated voice channel i hown in figure below. 4
5 TDM With TDM ytem, tranmiion from multiple ource occur on the ame tranmiion medium but not at the ame time. Tranmiion from variou ource i interleaved in time domain. Figure below how the timefrequency plan in TDM ytem, the ame communication reource i hared by aigning each of N ymbol or uer the full pectral occupancy of the ytem for a hort duration of time called time lot. The unued time region between lot aignment, called guard time, act a buffer zone to reduce interference. Figure below how a typical TDM ytem:  5
6 The multiplexing operation conit of providing each ource with an opportunity to occupy one or more lot. The demultiplexing operation conit of deloting the information and delivering the data to the intended ink. The communication witche (S 1 S M ) have ynchronized o that the maage correponding to ignal(1), for example, appear on the channel (1) output, and o on. Time i egmented in to interval called frame. Each frame i further partitioned in to aignable uer time lot. The implet TDM cheme called fixedaignment TDM. In fixed aignment TDM cheme, all of the lot ha no data to ent during a particular frame, that lot i wated. Another more efficient cheme, involving the dynamic aignment of the lot rather than fixed aignment. Figure below how the fixed aignment TDM ytem. Figure below how the fixed aignment and dynamic aignment TDM ytem. 6
7 Sampling Theorem The link between an analog waveform and it ampled verion i provided by what i known a the ampling proce. A band limited ignal having no pectral component above (f m Hz) can be determined uniquely by value ampled at uniform interval of T econd, where T 1 f m Stated another way, the upper limit on T can be expreed in term of the ampling rate, denoted f 1 T 7
8 The retriction, tated in term of ampling rate, i known a the Nyquit criterion. The tatement i The ampling rate ( f f m f f m ) alo called Nyquit rate. The Nyquit criterion i a theoretically ufficient condition to allow an analog ignal to be recontructed completely from a et of uniformly paced dicrete time ample. Impule Sampling Aume an analog waveform x(t), a hown in Fig. (a), with a Fourier tranform, X(f), which i zero outide the interval (f m <f<f m ), a hown in Fig. (b). The ampling of x(t) can be viewed a the product of x(t) with a train of unit impule function, x (t), hown in Fig. (c), and defined a follow: n x ( t) ( t nt ) Let u chooe T 1 f m, o that Nyquit rate i jut atified. Uing hifting property of the impule function the x (t), hown in Fig. (e), can be given by ( t) x( t) x ( t) x( t) ( t nt n x ) 8
9 n x ( nt ) ( t ) nt Uing frequency convolution property of Fourier tranform, the time product x( t) x ( t) tranform to the frequency domain convolution X ( f ) X ( f ), where ( f ) i the Fourier tranform of x (t) and given by X 1 X ( f ) ( f nf ) T n The convolution with an impule function imply hift the original function, a follow: given by: X( f ) ( f nf ) X( f nf ) The Fourier tranform of the ampled waveform, X (f), can be 1 X ( f ) X( f ) X ( f ) X ( f ) [ ( f nf )] T n 1 T n X ( f nf ) Figure below how the ampling theorem uing the frequency convolution property of the Fourier tranform (Impule ampling). 9
10 Natural Sampling In thi way the band limited analog ignal x(t), hown in Fig. (a 1 ), i multiplied by the pule train or witching waveform x p (t), hown in Fig. (c 1 ). Each pule in x p (t) ha width T and amplitude 1/T. The reulting ampled data equence, x (t), i hown in Fig. (e 1 ) and i expreed a x ( t) x( t) x ( t) The periodic pule train, x p (t), can be expreed a a Fourier erie in the form p 10
11 x p ( t) n c n e jnf t 1 nt and cn inc( ) T T where f =f m, T i the pule width, and 1/T i the pule amplitude. The envelope of magnitude pectrum of the pule train, een a a dahed line in Fig. (d 1 ), ha characteritic inc hape. x ( t) x( t) cn n e jnf t The Fourier tranform of x (t) i found a follow X ( f ) F[ x( t) cn n e jnf t For linear ytem the operation of ummation and Fourier tranformation can be interchanged. Therefore, X ( f ) cn n F[ x( t) e jnf t Uing frequency tranlation property of Fourier tranform, cn n X ( f ) X( f nf ) Note The ampling here i termed natural ampling, ince the top of each pule in the x (t) equence retain the hape of it correponding analog egment during the pule interval. ] ] 11
12 Figure below how the ampling theorem uing the hifting property of the Fourier tranform (Natural ampling). 1
13 Pule Modulation In pule modulation ome parameter of a pule train i varied in accordance with the maage ignal. Two familie of pule modulation may be ditinguihed: analog pule modulation and digital pule modulation. In analog pule modulation, a periodic pule train i ued a the carrier wave, and ome characteritic feature of each pule (e.g. Amplitude, Poition, and Width) i varied in a continuou manner in accordance with the correponding ample value of the meage ignal. Thu in analog pule modulation, information i tranmitted baically in analog form, but the tranmiion take place at dicrete time. In digital pule modulation, on the other hand, the maage ignal i repreented in a form that i dicrete in both time and amplitude; thereby permitting it tranmiion in digital form a a equence of coded pule. (1) Pule Amplitude Modulation (PAM) PAM i the implet and mot baic form of analog pule modulation. In PAM the amplitude of regularly paced pule are varied in proportion to the correponding ample value of a continuou meage ignal, the pule can be of a rectangular form or other appropriate hape. PAM a defined here i omewhat imilar to natural ampling where the meage ignal i multiplied by a periodic train of rectangular pule. However, in natural ampling the top of each 13
14 modulated rectangular pule varie with the meage ignal, wherea in PAM it i maintained flat. The waveform of PAM ignal i hown in figure below. There are two operation involved in the generation of the PAM ignal: (i) Intantaneou ampling of the meage ignal every T econd, where the ampling rate f =1/T i choen in accordance with the ampling theorem. (ii) Lengthening the duration of each ample o obtained to ome contant value ( ). 14
15 PAM/TDM Sytem Suppoe we wih to time multiplexed two ignal uing PAM. Let u aume that both input ignal f 1 (t) and f (t) are low pa, and band limited to 3KHz. The ampling theorem tate that each mut be ampled at a rate no le than 6KHz. Thi require a 1KHz minimum clock rate for the two channel ytem. Figure below how the block diagram of PAM/TDM ytem. The time multiplex PAM output might appear ome thing like that hown below. 15
16 The time pacing between adjacent ample in the time multiplex ignal waveform (T x ), can be defined a T x T n where T =ampling rate, and n=number of input ignal. To prevent any irretrievable lo of information in the compoite waveform then require that bandwidth B x of LPF mut atify the criterion B x 1 T x At the receiver the compoite time multiplexed and filtered waveform mut be reampled and eparated into the appropriate channel. One the pule are eparated, the normal ampling conideration applie and the analog recontruction of ignal can be obtained by LPF. The block diagram of PAM/TDM receiver i hown below. 16
17 Sample and Hold Circuit Figure below how the ample and hold circuit The witch cloe only when that particular channel i to be ampled. If the ource impedance r i mall, the capacitor voltage change to the input voltage within the time that witch i cloed. The load impedance R i arranged to be high o that the capacitor retain the voltage level until the witch i cloed again. Therefore the ample and hold circuit accept only thoe value of the input which occur at the ampling time and then hold them until the next ampling time. Example: Channel 1 of two channel PAM ytem handle 8KHz ignal. Channel handle 10 KHz ignal. The two channel are ampled at equal interval of time uing very narrow pule at the lowet frequency that i theoretical adequate. The ampled ignal are time multiplexed and paed through a LPF before tranmiion. 17
18 (1) What i the minimum clock frequency of the PAM ytem? () What i the minimum cut off frequency of LPF ued before tranmiion that will preerve the amplitude information on the output pule? (3) What would be the minimum bandwidth if thee channel were frequency multiplexed, uing AM technique and SSB technique? Solution (1) f 1 * f m1 f 1 *8 16KHz f f * f m *10 0KHz In order to ample channel adequately f f 0KHz The minimum clock rate n * f () T n T 50 Tx 5 ec n 1 Bx T B x 1 f 1 0KHz x 0KHz *0 50 ec 40KHz 18
19 (4) For AM For SSB min. BW. ( f m1 f m ) (8 10) 36KHz min. BW. f m1 f m KHz H.W Two low pa ignal, each band limited 4KHz, are to be time multiplexed into a ingle channel uing PAM. Each ignal i impule ampled at a rate 10KHz. The time multiplexed ignal waveform i filtered by an ideal LPF before tranmiion. (a) What i minimum clock frequency of the ytem? (b) What i the minimum cut off frequency of the LPF? (c) In the receiver ide, determine the minimum and maximum acceptable bandwidth of the LPF ued in retrieving the analog ignal? An. (a) 0KHz (b) 10KHz (c) 4KHz, 6KHz. 19
20 () Other Type of Analog Pule Modulation (PWM&PPM) One type of pule timing modulation ue contant amplitude pule whoe width i proportional to the value of meage ignal at the ampling intant. Thi type i deignated a pule width modulation (PWM) or pule duration modulation (PDM) i alo called. Another poibility i to keep both the amplitude and the width of the pule contant but vary the pule poition in proportion to the value of meage ignal at ampling intant. Thi i deignated a pule poition modulation (PPM). PAM, PWM and PPM waveform for a given meage ignal are hown below:  In PWM, the ignal f(t) i ampled periodically at a rate fat enough to atify the requirement of the ampling theorem. At each ampling intant a pule i generated with fixed amplitude and a width 0
21 that i proportional to the ample value of f(t). A minimum pule width i aigned to the minimum value of f(t). In PPM, thee are ent a contant width, contant amplitude pule. The minimum pule delay i ued to deignate the minimum value of f(t) and the change in delay i proportional to the modulating ignal. The contant of proportionality i the modulation contant. Generation of PWM & PPM Generation of PWM and PPM commonly employ variou combination of a ample and holed circuit, a preciion ramp voltage generator and a comparator. The block diagram of a typical circuit for generation PWM and PPM i hown in figure below:  1
22 The ramp generator produce a preciion ramp voltage which ha peak to peak amplitude lightly larger than the maximum amplitude range of the input ignal. Thi ramp voltage i the bai for the amplitude to timing converion and therefore mut be accurately known. The comparator i a high gain amplifier intended for two tated operation. If input ignal i higher than a preet reference level, the output i held in one tate (i.e. a given voltage level). Whenever the input ignal level i le than the reference level, the output i held in the other tate. Which output tate i preent, then, depend upon whether the input i above and below the threhold (reference level) of the comparator. The voltage reference level of the comparator i adjut o that there i alway an interection with the um of the ample and hold circuit and ramp voltage. In thi ytem, the firt croing of the reference level indicate the clock timing and the econd croing generate the variable trailing edge. A convenient way to generate PPM i to ue PWM waveform generated above and then trigger a contant width pule generation thoe edge of the PWM waveform with a negative lope.
23 Signal to noie ratio in analog pule modulation The performance of analog pule modulation ytem in the preence of additive noie i invetigated here. PAM Noie i added in the tranmiion of the PAM ignal a illutration in figure below. The noie occurring between pule add noie power to the tranmiion without any increae in ignal power. To avoid thi, a ynchronized gating circuit i ued in the receiver to accept ample only when the ignal i known to be preent. We hall aume that the ignal and the additive noie preent in the input to the PAM receiver are band limited and that the condition of the ampling theorem are atified. Becaue the PAM receiver i linear, we can apply the ignal and the noie eparately meaure their power, and then combine. The ampling and low pa filtering at the receiver reproduce the band limited ignal and noie pectra within a contant, a hown in figure below. 3
24 Thu S ( t) K S ( ) 0 i t n0 ( t) K n ( t) i So that S N 0 0 S N i i Pule timing modulation Although the pule to convey the information may be generated with extremely hort (fat) rie time, after paing through a band limited ytem they have rie time which are governed by the bandwidth of the ytem. Thi rie time can be approximated by a linear ramp. A hown in figure below, o that the pule aume a trapezoidal hape. The poition of the trapezoidal pule i enitive to additive noie. If the noie voltage i aumed to vary lowly compared to the rie time of the pule, the variation in the pule amplitude, n, may be 4
25 repreented by a hift,, in the pule poition a hown in above figure. From the geometry of above figure, we have t r n A or t ( r ) n (1) A The output ignal amplitude i proportional to the modulating ignal f(t) through a modulation contant k. or S0 ( t) kf ( t) S ( t) k f ( ) () 0 t The output noie ue Eq.(1) Alo we have n S tr ( ) ni ( ) (3) A 0 t i A ( ) (4) T and for ideal LPF (giving a nearly linear rie time) B 1 (5) t r 5
26 Combining Eq. () and (5) we have S0 N 0 K f ( t) B / T S N i i S 0 N 0 B Si N i the S/N improvement in a PPM i proportional to the quare of the bandwidth. Pule code modulation (PCM) Pule code modulation (PCM) i the name given to the cla of baeband ignal obtained from the quantized PAM ignal by encoding each quantized ample into a digital word. Figure below how the tep required in PCM communication. Input ignal Sampler PAM ignal Quantizer Encoder PCM ignal The ource of information i ampled and quantized to one of L level, then each quantized ample i digitally encoded into a kbit code word. Where k log k L L The eential feature of binary PCM are hown in figure below. Aume that an analog ignal, x(t), i limited in it excurion to the 6
27 range (4V to +4V). The tep ize between quantization level ha been et at 1V. Thu eight quantization level are employed, thee located at 3.5V, .5V,., +3.5V. The code number 0 may be aigned to the level at 3.5V; the code number 1 may be aigned to the level at .5V, and o on until the level at 3.5V, which i aigned the code number 7. Each code number ha it repreentation in binary arithmetic, ranging from 000 for code number 0 to 111 for code number 7. 7
28 From the above figure each ample of analog ignal i aigned to the quantization level cloet to the value of the ample. Beneath the analog waveform, x(t), are een four repreentation of x(t) a follow: the natural ample value, the quantized ample value, the code number, and the PCM equence. Quantization The objective of the quantization tep in PCM proce i to repreent each ample by a fixed number of bit. For example, if the amplitude of PAM reulting from ampling proce range between (1V and +1V), there can be infinite value of voltage between (1 and +1). For intance, one value can be V. To aign a different binary equence to each voltage value, we would have to contruct a code of infinite length. Therefore, we can take a limit number of voltage value between (1V and +1V) to repreent the original ignal and thee value mut be dicrete. Aume that the quantization tep were in 0.1V increment, and the voltage meaurement for one ample i 0,58V. That would have to be rounded off to 0.6V, the nearet dicrete value. Note that there i a 0.0V error, the difference between 0.58V and 0.6V. See figure below. Take tep 1 in the curve, for example, the curve i paing through a maximum and i given tow value of 1. For the firt value, the actual curve i above 1 and for econd value below 1. That error from the true value to the quantum value i called quantization ditortion. Thi ditortion i the major ource of imperfection in PCM ytem. 8
29 Code value The more quantization level, the better quality the ytem will deliver. However, increaing the number of quantization level ha two major cot: 1) The cot of deigning a ytem with large binary code ize needed. ) The time it take to proce thi large number of quantizing tep by the coder. Therefore, a very large number of quantizing level may induce unwanted delay in the ytem. 9
30 Uniform and Nonuniform Quantization Form the above dicuion it cam be een that the quantization noie depend on the tep ize. When the tep have uniform ize the quantization called a uniform quantization. For uniform quantization, the quantization noie i the ame for all ignal magnitude. Therefore, with uniform quantization the ignal to noie ratio (SNR) i wore for low level ignal than for high level ignal. Nonuniform quantization can provide fine quantization of the weak ignal and coare quantization of the trong ignal. Thu in the cae of nonuniform quantization, quantization noie can be made proportional to ignal ize. The effect i to improve the overall SNR by reducing the noie for the predominant weak ignal, at the expene of an increae in noie for the rarely occurring trong ignal. Figure below compare the quantization of trong ignal veru a weak ignal for uniform and nonuniform quantization. 30
31 Encoding Figure below ome of the more commonly ued PCM repreentation. 1) Returntozero (RZ) method repreent a 1 by a change to the level for onehalf the bit interval, after which the ignal return to the reference level for the remaining halfbit interval. A 0 i indicated in thi method by no change, the ignal remaining at the reference level. 31
32 ) Returntobia (RB) method, in thi method three level are ued 0, 1, and a bia level. The bia level may be choen either below or between the other two level. The waveform return to the bia level during the lat half of each bit interval. 3) Alternate Mark Inverion (AMI), in thi method the firt binary one i repreented by +1, the econd by 1, the third by +1, etc. The AMI repreentation i eaily derived from an RZ binary code (and vice vera) by alternately inverting the 1. It ha zero average value and i widely ued in telephone PCM ytem. Thi i alo referred to a a bipolar returntozero (BRZ) repreentation. 4) Splitphae repreentation eliminate the variation in average value uing ymmetry. In the Mancheter plitphae method, a 1 i repreented by a 1 level during the firt halfbit interval, then hifted to the 0 level for the latter halfbit interval; a 0 i indicated by the revere repreentation. In the plitphae (mark) method, a imilar ymmetric repreentation i ued expect that a phae reveral relative to the previou phae indicate a 1 (i.e. mark) and no change in phae i ued to indicate a 0. 5) Nonreturntozero (NRZ) repreentation reduce the bandwidth needed to end PCM code. In NRZ (L) repreentation a bit pule remain in one of it two level for the entire bit interval. In NRZ 3
33 (M) method a level change i ued to indicate a mark (i.e a 1) and no level change for a 0; the NRZ (S) method ue the ame cheme except that a level change i ued to indicate a pace (i.e. a 0). Both of thee are example of the more general claification NRZ (I) in which a level change (inverion) i ued to indicate one kind of binary digit and no level change indicate the other digit. Note that ue of plitphae and NRZ repreentation require ome added receiver complexity to determine the clock frequency. 6) Delay modulation (Miller code), in thi method a 1 i repreented by a ignal tranition at the midpoint of a bit interval. A 0 i repreented by no tranition unle it i followed by another 0, in which cae the ignal tranition occur at the end of the bit interval. In thi method, a ucceion of 1 and a ucceion of 0 each are repreented by a quare wave at the bit rate, but one i delayed a halfbit interval from the other. Noie conideration in PCM ytem The performance of a PCM ytem i influenced by two major ource of noie. 1) Channel noie, which i introduced anywhere between the tranmitter output and the receiver input, channel noie i alway preent, once the equipment i witched on. 33
34 ) Quantization noie, which i introduced in the tranmitter and i carried all the way along to the receiver output. Quantization Noie The peak ignal to r.m. noie power ratio i given by S N 0 0 3L S0 ) 4.8 0log N 0 db 10 where L=number of quantizer level. S 0 = peak ignal power. N 0 = r.m. noie power. L Increaing L increae the number of code pule and hence the bandwidth. We can thu relate SNR to bandwidth. Thi i eaily done by noting that m L n where m=the number pule in code group. n=the number of code level. and N S 0 m 3n 0 S0 ) 4.8 0mlog N 0 db 10 n 34
35 In particular, for binary code n=. S N 0 ) db m Since the bandwidth i proportional to m, the output SNR increae exponentially with bandwidth. Interymbol interference (ISI) and pule haping to reduce ISI Conider the equence of pule hown in figure below. Although thee are hown a binary pule, they could well be pule of identical hape, but of arbitrary height. They are hown recurring at T econd, where T i the ampling interval. Sytem filtering caue thee pule to pread out a they travere the ytem. And they overlap into adjacent time lot a hown. At the receiver the original pule meage may be derived by ampling at the center of each time lot, and then baing a deciion on the amplitude of the ignal meaured at that point. 35
36 The ignal overlapped into adjacent time lot may, if too trong, reult in an erroneou deciion. Thu, a an example, in the cae of above figure the 0 tranmitted may appear a a 1 if tail of adjacent pule add up to too high a value. Thi phenomenon of pule overlap and the reultant difficulty with receiver deciion i termed interymbol interference (ISI). Nyquit invetigated the problem of pecifying a received pule hape o that no ISI occur at the detector. He howed that the theoretical minimum ytem bandwidth needed to detect R ymbol/econd without ISI i R / Hz. Thi occur when the ytem tranfer function, H(f), i made rectangular a hown in figure below. 36
37 Note that, when H(f) i uch an ideal filter with bandwidth 1/T, it impule repone 1/T Hz. t h( t) in c( ) T Therefore, the bandwidth required to detect 1/T ymbol/ec. i The Nyquit pule hape i not phyically realizable ince it dictate a rectangular bandwidth characteritic. Alo, with uch a characteritic, the detection proce would be very enitive to mall timing error. One frequently ued ytem tranfer function H(f) i called the raied coine filter. It can be expreed a H ( f ) 1 1 [(T [1 co( 0 f ) 1 r] )] r 1 r 0 f T 1 r 1 r f T T 1 r f T where 0 r i called rolloff factor. 0 = abolute bandwidth. 0 1 T =minimum Nyquit bandwidth. 37
38 Therefore, the bandwidth may be given by: (1 r) BW. T Note: If r 0 0, i the Nyquit minimum bandwidth. If r T The correponding impule repone for the H(f) of raied coine filter i: t rt in co h( t) ( ) ( T t T 4rt 1 ( ) T T ) Figure below how the raied coine filter characteritic. 38
39 39
40 Channel Capacity The maximum rate of tranmiion wa found by Shannon to be given by: C W (1 log S N Shannon maximum capacity expreion provide an upper bound on the rate at which one can communicate over a channel of bandwidth W, and ignal to noie ratio SNR ( N S ). Delta modulation (DM) ) In the baic form, DM provide a tair cae approximation to the over ampled verion of the meage ignal, a hown in figure below, the difference between the input and the approximation i quantized into only two level, namely,, correponding to poitive and negative difference, repectively. Thu, if the approximation below the ignal at any ampling epoch, it i increaed by, on the other hand, the approximation lie above the ignal, it i diminihed by. 40
41 Denoting the input ignal a m(t), and it tair cae approximation a m q (t), the baic principle of DM may be formalized in the following et of dicrete time relation: e e ( nt q ) m( nt ) m ( nt T ) (a) q ( nt m ) Sgn[ e( nt )] (b) q ( nt q q ) m ( nt T ) e ( nt ) (c) where T =ampling time, e(nt )=error ignal repreenting difference between m(nt ) and the latet approximation to it m(nt )m q (nt T ), e q (nt )= the quantized verion of e(nt ). Finally, the quantizer output e q (nt ) coded to produce the DM ignal. DM modulator and demodulator are hown below. 41
42 The comparator compute the difference between two input. The quantizer conit of hard limiter with an input/output relation that i caled verion of the ignum function. The accumulator increment the approximation by a tep in poitive or negative direction, depending on the algebraic ign of the error ignal e(nt ). Demodulation i ubjected to two type of error: (1) Slop over load direction. () Granular noie. Equation (c) may be oberved a a digital equivalent of integration in the ene that it repreent the accumulation of poitive and negative increment of magnitude, alo, denoting the quantization error by q(nt ), a hown by m q ( nt ) m( nt ) q( nt ) From equation (a) may be oberved e( nt ) m( nt ) m( nt T ) q( nt T ) Thu, except for the quantization error q(nt T ), the quantizer input i a firt backward difference of the input ignal, which may be viewed a a digital approximation to the derivative of the input ignal or, equivalently, a the invere of the digital integration proce. Conider the maximum lope of input m(t), it i clear that in order for the equence of ample m q (nt ) to increae a fat a the input equence of ample m(nt ) in a region of maximum lope of m(t), the condition 4
43 dm( t) max T dt mut be atified. The tep ize ( ) i too mall for the taircae approximation m q (t) to follow a teep egment of the input ignal m(t), with the reult that m q (t) fall behind m(t), a hown in figure below. Thi condition called lope overload, and the reulting quantization noie called lope overload ditortion. Delta modulation uing a fixed tep ize ( ) i often referred to a a linear delta modulator. In contrat to lopoverload ditortion, granular noie occur when the tep ize ( ) i too large relative to the local lope characteritic of the input waveform m(t), there by cauing the taircae approximation m q (t) to hunt around a relatively flat egment of the input waveform. Thi noie alo illutrated in above figure. 43
44 It i therefore clear that the choice of optimum tep ize that minimize the mean quare value of quantization error in a linear delta modulator will be the reult of a compromie between lope overload ditortion and granular ditortion. To atify uch a requirement, the modulator mut be made (adaptive) in the ene that the tep ize i made to vary in accordance with the input ignal. Adaptive delta modulation Adaptive delta modulation i a delta modulation where the tep ize ( ) i automatically varied depending on the amplitude characteritic of the analog input ignal, a hown in figure below. 44
45 Noie in communication ytem There i one natural ource of noie called thermal or Johnon noie that can not be eliminated. Thermal noie caued by thermal motion of electron in all diipative component. Thermal noie can be decribed a a zero mean Gauian random proce. A Gauian proce, n(t), i a random function whoe value, n, at any arbitrary time, t, i tatitically characterized by the Gauian probability denity function, P(n): P( n) where =variance of n. 1 1 n exp[ ( ) ] If =1 called normalized probability denity function, a hown in figure below. Gauian ditribution i often ued a a ytem noie model becaue of a theorem called the central limit theorem, which tate that under very 45
46 general condition the probability ditribution of the um of j tatitically independent random variable approache the Gauian ditribution a j. White noie The primary pectral characteritic of thermal noie i that it pectral denity i the ame for all frequencie of interet in mot communication ytem. Therefore, a imple model for thermal noie aume that it power pectral denity G n (f) i flat for all frequencie, a hown in figure below, and i denoted a follow N0 G n ( f ) watt/hz The factor i included to indicated that G n (f) i a two ided power pectral denity. When noie power ha uch a uniform pectral denity, it i called white noie. G n (f) N 0 / f 0 The autocorrelation function of t white noie i given by: R n 1 N0 ( ) F [ Gn( f )] ( ) R n ( ) N 0 / 0 t 46
47 The average power, P n, of white noie = becaue of bandwidth=. N P n 0 df Since thermal noie i preent in all communication ytem, the thermal noie characteritic additive, white, and Gauian are mot often ued to model the noie in communication ytem, and called a Additive White Gauian Noie (AWGN). Example 1 In the pule timing modulation receiver the ratio of peak amplitude to additive r.m. noie=10, the pule duration=1 ec., the guard time= 1 ec., the minimum bandwidth=3.3 MHz, and the SNR before demodulation=10. (a) Find SNR after demodulation. (b) If the amplitude rang 1volt, calculate the ignal reolution. Solution: Si N i A n ( ) T ( t) T T mod T 6 T 10 ec. g 10 8 ec. 47
48 S0 k f ( t) B N 0 T Si N i k f mod V m ( t)( ) Si N T k k=modulation contant. k V m T mod k Tmod 4Vm Tmod k Vm i for in wave f ( t) Vm k S N ( t) T ( mod 1 f ) 3 10 ec (b) k 410 ec. t r n t ( A 1 ( ) ) B 1 t r t r 0.15 ec. 48
49 k reolution (0.01) ec. 6 reolution ec. 3. V k 4 10 ec./ V 75 6 Example The information in an analog waveform, with maximum frequency f m =3 KHz, i to be tranmitted over M level PCM ytem, where number of pule level M=16. The quantization ditortion i pecified not to exceed 1% of the peaktopeak analog ignal. (a)what i the minimum number of bit/ample, or bit/pcm word that hould be ued in thi PCM ytem. (b)what i the minimum required ampling rate, and what i the reulting bit tranmiion rate. (c) What i the PCM pule or ymbol tranmiion rate. Solution: Note: in thi example we are conidered with two type of level, the number of quantization level (L), and the 16 level of the multilevel PCM pule (M). (a) By uing 1 L level p 49
50 1 k log bit p where L=number of quantization level, k=number of bit, and p=fraction of peaktopeak analog voltage. 1 k log log k k 6 L 64 The number of bit/ample =k =6 5.6 (b) f =f m =6000 ample/econd bit tranmiion rate R kf b R b bit/ec. (c) ince multilevel pule are to be ued with M= m =16 m=4 bit/ymbol The bit tream will be partitioned into group of 4bit to form a new 16level PCM digit. Symbol tranmiion rate (R ) R Rb 36000bit / ec. ymbol 9000 m 4bit / ymbol ec. 50
51 Example 3 (a) Find the minimum required bandwidth for the baeband tranmiion of 4level PCM pule equence having a data rate of R b =400 bit/ec. if the ytem tranfer characteritic conit of a raied coine pectrum with 100% exce bandwidth (r=1). (b) The ame PCM equence i modulated on to a carrier wave, o that the baeband pectrum i hifted and centered at frequency f 0. Find the minimum required DSB bandwidth for tranmitting the modulated PCM equence. Aume that the ytem tranfer characteritic i the ame in part (a). Solution: (a) M= m, ince M=4, m= bit/ymbol Pule or ymbol rate R Rb 400bit / ec. ymbol 100 m bit / ymbol ec. Minimum bandwidth (1 r) R (1 1) Hz (b) WDSB ( 1 r) R (1 1) Hz 51
52 Example 4 Ten voice channel each of bandwidth (B.W) =3. KHz are equentially ampled at 8 KHz and TDM ed. (a) What i the ytem bandwidth (B.W). (b) If TDM ed ignal i PCM ed uing 8level quantization, find bit rate (R b ) Solution: (a)without guard band T 1 f 1 8KHz 15 ec. 10 voice channel, 10 ample 1 Neceary B.W= 1510 (b) k log L log bit ample 80KHz R b 3 ample bit bit ec. ample ec. Example 5 A Delta modulator i ued to encode peech ignal bandlimited to 3KHz with ampling frequency 100 KHz. For 1 volt peak ignal voltage, find (a) Minimum tep ize to avoid lope overloading. 5
53 (b) Signal to quantization noie ratio if peech i aumed to have nonuniform probability denity function (PDF). Solution: For DM ytem, if input ignal f ( t) b co t. m (a) df ( t) dt max b f if tep ize ued in DM ytem=a f a f m a / b f f m b m a V 3 minimum tep ize (b) S N f 3 ( ) fm ( ) 53
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